请输入您要查询的字词:

 

单词 AbelianGroup
释义

abelian group


Let (G,*) be a group. If for any a,bG we havea*b=b*a, we say that the group is abelianMathworldPlanetmath (or commutativePlanetmathPlanetmathPlanetmath).Abelian groups are named after Niels Henrik Abel, but the word abelian is commonly written in lowercase.

Abelian groups are essentially the same thing as unitary -modules (http://planetmath.org/Module).In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using additive notation.

Some of the basic properties of abelian groups are as follows:

Theorem 1.

Any subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) of an abelian group is normal.

Proof.

Let H be a subgroup of the abelian group G. Since ah=ha for any aG and any hH we get aH=Ha. That is, H is normal in G.∎

Theorem 2.

Quotient groupsMathworldPlanetmath of abelian groups are also abelian.

Proof.

Let H be a subgroup of G. Since G is abelian, H is normal and we can get the quotient group G/H whose elements are the equivalence classesMathworldPlanetmathPlanetmath forab if ab-1H.The operationMathworldPlanetmath on the quotient group is given by aHbH=(ab)H. But bHaH=(ba)H=(ab)H, therefore the quotient group is also commutative.∎

Here is another theorem concerning abelian groups:

Theorem 3.

If φ:GG defined by φ(x)=x2 is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/GroupHomomorphism), then G is abelian.

Proof.

If such a function were a homomorphism,we would have

(xy)2=φ(xy)=φ(x)φ(y)=x2y2

that is, xyxy=xxyy.Left-multiplying by x-1 and right-multiplying by y-1 we are led toyx=xy and thus the group is abelian.∎

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 9:19:08